Sr Examen

Gráfico de la función y = sinx/|x|

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src]
       sin(x)
f(x) = ------
        |x|  
f(x)=sin(x)xf{\left(x \right)} = \frac{\sin{\left(x \right)}}{\left|{x}\right|}
f = sin(x)/|x|
Gráfico de la función
02468-8-6-4-2-10102-2
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
sin(x)x=0\frac{\sin{\left(x \right)}}{\left|{x}\right|} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=πx_{1} = \pi
Solución numérica
x1=72.2566310325652x_{1} = 72.2566310325652
x2=59.6902604182061x_{2} = -59.6902604182061
x3=3.14159265358979x_{3} = 3.14159265358979
x4=43.9822971502571x_{4} = -43.9822971502571
x5=370.707933123596x_{5} = -370.707933123596
x6=81.6814089933346x_{6} = 81.6814089933346
x7=100.530964914873x_{7} = -100.530964914873
x8=28.2743338823081x_{8} = 28.2743338823081
x9=65.9734457253857x_{9} = 65.9734457253857
x10=31.4159265358979x_{10} = -31.4159265358979
x11=9.42477796076938x_{11} = -9.42477796076938
x12=40.8407044966673x_{12} = 40.8407044966673
x13=56.5486677646163x_{13} = 56.5486677646163
x14=56.5486677646163x_{14} = -56.5486677646163
x15=12.5663706143592x_{15} = 12.5663706143592
x16=43.9822971502571x_{16} = 43.9822971502571
x17=100.530964914873x_{17} = 100.530964914873
x18=3.14159265358979x_{18} = -3.14159265358979
x19=15.707963267949x_{19} = -15.707963267949
x20=59.6902604182061x_{20} = 59.6902604182061
x21=6.28318530717959x_{21} = 6.28318530717959
x22=9.42477796076938x_{22} = 9.42477796076938
x23=53.4070751110265x_{23} = -53.4070751110265
x24=47.1238898038469x_{24} = -47.1238898038469
x25=87.9645943005142x_{25} = -87.9645943005142
x26=69.1150383789755x_{26} = 69.1150383789755
x27=21.9911485751286x_{27} = 21.9911485751286
x28=87.9645943005142x_{28} = 87.9645943005142
x29=18.8495559215388x_{29} = 18.8495559215388
x30=84.8230016469244x_{30} = -84.8230016469244
x31=72.2566310325652x_{31} = -72.2566310325652
x32=270.176968208722x_{32} = 270.176968208722
x33=37.6991118430775x_{33} = 37.6991118430775
x34=25.1327412287183x_{34} = 25.1327412287183
x35=50.2654824574367x_{35} = 50.2654824574367
x36=34.5575191894877x_{36} = 34.5575191894877
x37=6.28318530717959x_{37} = -6.28318530717959
x38=65.9734457253857x_{38} = -65.9734457253857
x39=21.9911485751286x_{39} = -21.9911485751286
x40=62.8318530717959x_{40} = -62.8318530717959
x41=75.398223686155x_{41} = 75.398223686155
x42=84.8230016469244x_{42} = 84.8230016469244
x43=53.4070751110265x_{43} = 53.4070751110265
x44=15.707963267949x_{44} = 15.707963267949
x45=28.2743338823081x_{45} = -28.2743338823081
x46=25.1327412287183x_{46} = -25.1327412287183
x47=91.106186954104x_{47} = -91.106186954104
x48=47.1238898038469x_{48} = 47.1238898038469
x49=97.3893722612836x_{49} = 97.3893722612836
x50=69.1150383789755x_{50} = -69.1150383789755
x51=94.2477796076938x_{51} = 94.2477796076938
x52=18.8495559215388x_{52} = -18.8495559215388
x53=50.2654824574367x_{53} = -50.2654824574367
x54=37.6991118430775x_{54} = -37.6991118430775
x55=376.991118430775x_{55} = 376.991118430775
x56=81.6814089933346x_{56} = -81.6814089933346
x57=62.8318530717959x_{57} = 62.8318530717959
x58=78.5398163397448x_{58} = 78.5398163397448
x59=153.9380400259x_{59} = 153.9380400259
x60=31.4159265358979x_{60} = 31.4159265358979
x61=78.5398163397448x_{61} = -78.5398163397448
x62=40.8407044966673x_{62} = -40.8407044966673
x63=223.053078404875x_{63} = -223.053078404875
x64=97.3893722612836x_{64} = -97.3893722612836
x65=113.097335529233x_{65} = -113.097335529233
x66=75.398223686155x_{66} = -75.398223686155
x67=91.106186954104x_{67} = 91.106186954104
x68=590.619418874881x_{68} = 590.619418874881
x69=12.5663706143592x_{69} = -12.5663706143592
x70=94.2477796076938x_{70} = -94.2477796076938
x71=34.5575191894877x_{71} = -34.5575191894877
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en sin(x)/|x|.
sin(0)0\frac{\sin{\left(0 \right)}}{\left|{0}\right|}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
cos(x)xsin(x)sign(x)x2=0\frac{\cos{\left(x \right)}}{\left|{x}\right|} - \frac{\sin{\left(x \right)} \operatorname{sign}{\left(x \right)}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=86.3822220347287x_{1} = -86.3822220347287
x2=4.49340945790906x_{2} = -4.49340945790906
x3=42.3879135681319x_{3} = -42.3879135681319
x4=32.9563890398225x_{4} = -32.9563890398225
x5=67.5294347771441x_{5} = -67.5294347771441
x6=20.3713029592876x_{6} = -20.3713029592876
x7=23.519452498689x_{7} = 23.519452498689
x8=61.2447302603744x_{8} = 61.2447302603744
x9=7.72525183693771x_{9} = 7.72525183693771
x10=92.6661922776228x_{10} = -92.6661922776228
x11=14.0661939128315x_{11} = 14.0661939128315
x12=45.5311340139913x_{12} = -45.5311340139913
x13=48.6741442319544x_{13} = 48.6741442319544
x14=70.6716857116195x_{14} = 70.6716857116195
x15=64.3871195905574x_{15} = 64.3871195905574
x16=95.8081387868617x_{16} = 95.8081387868617
x17=36.1006222443756x_{17} = 36.1006222443756
x18=29.811598790893x_{18} = -29.811598790893
x19=76.9560263103312x_{19} = -76.9560263103312
x20=10.9041216594289x_{20} = -10.9041216594289
x21=98.9500628243319x_{21} = 98.9500628243319
x22=76.9560263103312x_{22} = 76.9560263103312
x23=45.5311340139913x_{23} = 45.5311340139913
x24=39.2444323611642x_{24} = 39.2444323611642
x25=98.9500628243319x_{25} = -98.9500628243319
x26=89.5242209304172x_{26} = -89.5242209304172
x27=61.2447302603744x_{27} = -61.2447302603744
x28=17.2207552719308x_{28} = 17.2207552719308
x29=92.6661922776228x_{29} = 92.6661922776228
x30=4.49340945790906x_{30} = 4.49340945790906
x31=54.9596782878889x_{31} = 54.9596782878889
x32=394.267341680887x_{32} = -394.267341680887
x33=48.6741442319544x_{33} = -48.6741442319544
x34=67.5294347771441x_{34} = 67.5294347771441
x35=7.72525183693771x_{35} = -7.72525183693771
x36=17.2207552719308x_{36} = -17.2207552719308
x37=86.3822220347287x_{37} = 86.3822220347287
x38=32.9563890398225x_{38} = 32.9563890398225
x39=26.6660542588127x_{39} = -26.6660542588127
x40=26.6660542588127x_{40} = 26.6660542588127
x41=80.0981286289451x_{41} = 80.0981286289451
x42=108.375719651675x_{42} = 108.375719651675
x43=95.8081387868617x_{43} = -95.8081387868617
x44=20.3713029592876x_{44} = 20.3713029592876
x45=83.2401924707234x_{45} = -83.2401924707234
x46=10.9041216594289x_{46} = 10.9041216594289
x47=83.2401924707234x_{47} = 83.2401924707234
x48=202.627791039417x_{48} = -202.627791039417
x49=89.5242209304172x_{49} = 89.5242209304172
x50=29.811598790893x_{50} = 29.811598790893
x51=58.1022547544956x_{51} = 58.1022547544956
x52=54.9596782878889x_{52} = -54.9596782878889
x53=64.3871195905574x_{53} = -64.3871195905574
x54=39.2444323611642x_{54} = -39.2444323611642
x55=14.0661939128315x_{55} = -14.0661939128315
x56=70.6716857116195x_{56} = -70.6716857116195
x57=73.8138806006806x_{57} = -73.8138806006806
x58=73.8138806006806x_{58} = 73.8138806006806
x59=36.1006222443756x_{59} = -36.1006222443756
x60=58.1022547544956x_{60} = -58.1022547544956
x61=42.3879135681319x_{61} = 42.3879135681319
x62=51.8169824872797x_{62} = -51.8169824872797
x63=23.519452498689x_{63} = -23.519452498689
x64=51.8169824872797x_{64} = 51.8169824872797
x65=80.0981286289451x_{65} = -80.0981286289451
Signos de extremos en los puntos:
(-86.38222203472871, 0.0115756804584678)

(-4.493409457909064, 0.217233628211222)

(-42.38791356813192, 0.0235850682290164)

(-32.956389039822476, -0.0303291711863103)

(-67.52943477714412, 0.0148067339465492)

(-20.37130295928756, -0.0490296240140742)

(23.519452498689006, -0.0424796169776126)

(61.2447302603744, -0.0163257593209978)

(7.725251836937707, 0.128374553525899)

(-92.66619227762284, 0.0107907938495342)

(14.066193912831473, 0.0709134594504622)

(-45.53113401399128, -0.0219576982284824)

(48.674144231954386, -0.0205404540417537)

(70.6716857116195, 0.0141485220648664)

(64.38711959055742, 0.0155291838074613)

(95.8081387868617, 0.0104369581345658)

(36.10062224437561, -0.0276897323011492)

(-29.81159879089296, 0.0335251350213988)

(-76.95602631033118, -0.0129933369870427)

(-10.904121659428899, 0.0913252028230577)

(98.95006282433188, -0.010105591736504)

(76.95602631033118, 0.0129933369870427)

(45.53113401399128, 0.0219576982284824)

(39.24443236116419, 0.0254730530928808)

(-98.95006282433188, 0.010105591736504)

(-89.52422093041719, -0.0111694646341736)

(-61.2447302603744, 0.0163257593209978)

(17.22075527193077, -0.0579718023461539)

(92.66619227762284, -0.0107907938495342)

(4.493409457909064, -0.217233628211222)

(54.959678287888934, -0.0181921463218031)

(-394.26734168088706, 0.00253634191261283)

(-48.674144231954386, 0.0205404540417537)

(67.52943477714412, -0.0148067339465492)

(-7.725251836937707, -0.128374553525899)

(-17.22075527193077, 0.0579718023461539)

(86.38222203472871, -0.0115756804584678)

(32.956389039822476, 0.0303291711863103)

(-26.666054258812675, -0.0374745199939312)

(26.666054258812675, 0.0374745199939312)

(80.09812862894512, -0.012483713321779)

(108.37571965167469, 0.00922676625078197)

(-95.8081387868617, -0.0104369581345658)

(20.37130295928756, 0.0490296240140742)

(-83.2401924707234, -0.0120125604820527)

(10.904121659428899, -0.0913252028230577)

(83.2401924707234, 0.0120125604820527)

(-202.62779103941682, -0.00493509709208483)

(89.52422093041719, 0.0111694646341736)

(29.81159879089296, -0.0335251350213988)

(58.10225475449559, 0.0172084874716279)

(-54.959678287888934, 0.0181921463218031)

(-64.38711959055742, -0.0155291838074613)

(-39.24443236116419, -0.0254730530928808)

(-14.066193912831473, -0.0709134594504622)

(-70.6716857116195, -0.0141485220648664)

(-73.81388060068065, 0.01354634434514)

(73.81388060068065, -0.01354634434514)

(-36.10062224437561, 0.0276897323011492)

(-58.10225475449559, -0.0172084874716279)

(42.38791356813192, -0.0235850682290164)

(-51.81698248727967, -0.019295099487588)

(-23.519452498689006, 0.0424796169776126)

(51.81698248727967, 0.019295099487588)

(-80.09812862894512, 0.012483713321779)


Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
x1=32.9563890398225x_{1} = -32.9563890398225
x2=20.3713029592876x_{2} = -20.3713029592876
x3=23.519452498689x_{3} = 23.519452498689
x4=61.2447302603744x_{4} = 61.2447302603744
x5=45.5311340139913x_{5} = -45.5311340139913
x6=48.6741442319544x_{6} = 48.6741442319544
x7=36.1006222443756x_{7} = 36.1006222443756
x8=76.9560263103312x_{8} = -76.9560263103312
x9=98.9500628243319x_{9} = 98.9500628243319
x10=89.5242209304172x_{10} = -89.5242209304172
x11=17.2207552719308x_{11} = 17.2207552719308
x12=92.6661922776228x_{12} = 92.6661922776228
x13=4.49340945790906x_{13} = 4.49340945790906
x14=54.9596782878889x_{14} = 54.9596782878889
x15=67.5294347771441x_{15} = 67.5294347771441
x16=7.72525183693771x_{16} = -7.72525183693771
x17=86.3822220347287x_{17} = 86.3822220347287
x18=26.6660542588127x_{18} = -26.6660542588127
x19=80.0981286289451x_{19} = 80.0981286289451
x20=95.8081387868617x_{20} = -95.8081387868617
x21=83.2401924707234x_{21} = -83.2401924707234
x22=10.9041216594289x_{22} = 10.9041216594289
x23=202.627791039417x_{23} = -202.627791039417
x24=29.811598790893x_{24} = 29.811598790893
x25=64.3871195905574x_{25} = -64.3871195905574
x26=39.2444323611642x_{26} = -39.2444323611642
x27=14.0661939128315x_{27} = -14.0661939128315
x28=70.6716857116195x_{28} = -70.6716857116195
x29=73.8138806006806x_{29} = 73.8138806006806
x30=58.1022547544956x_{30} = -58.1022547544956
x31=42.3879135681319x_{31} = 42.3879135681319
x32=51.8169824872797x_{32} = -51.8169824872797
Puntos máximos de la función:
x32=86.3822220347287x_{32} = -86.3822220347287
x32=4.49340945790906x_{32} = -4.49340945790906
x32=42.3879135681319x_{32} = -42.3879135681319
x32=67.5294347771441x_{32} = -67.5294347771441
x32=7.72525183693771x_{32} = 7.72525183693771
x32=92.6661922776228x_{32} = -92.6661922776228
x32=14.0661939128315x_{32} = 14.0661939128315
x32=70.6716857116195x_{32} = 70.6716857116195
x32=64.3871195905574x_{32} = 64.3871195905574
x32=95.8081387868617x_{32} = 95.8081387868617
x32=29.811598790893x_{32} = -29.811598790893
x32=10.9041216594289x_{32} = -10.9041216594289
x32=76.9560263103312x_{32} = 76.9560263103312
x32=45.5311340139913x_{32} = 45.5311340139913
x32=39.2444323611642x_{32} = 39.2444323611642
x32=98.9500628243319x_{32} = -98.9500628243319
x32=61.2447302603744x_{32} = -61.2447302603744
x32=394.267341680887x_{32} = -394.267341680887
x32=48.6741442319544x_{32} = -48.6741442319544
x32=17.2207552719308x_{32} = -17.2207552719308
x32=32.9563890398225x_{32} = 32.9563890398225
x32=26.6660542588127x_{32} = 26.6660542588127
x32=108.375719651675x_{32} = 108.375719651675
x32=20.3713029592876x_{32} = 20.3713029592876
x32=83.2401924707234x_{32} = 83.2401924707234
x32=89.5242209304172x_{32} = 89.5242209304172
x32=58.1022547544956x_{32} = 58.1022547544956
x32=54.9596782878889x_{32} = -54.9596782878889
x32=73.8138806006806x_{32} = -73.8138806006806
x32=36.1006222443756x_{32} = -36.1006222443756
x32=23.519452498689x_{32} = -23.519452498689
x32=51.8169824872797x_{32} = 51.8169824872797
x32=80.0981286289451x_{32} = -80.0981286289451
Decrece en los intervalos
[98.9500628243319,)\left[98.9500628243319, \infty\right)
Crece en los intervalos
(,202.627791039417]\left(-\infty, -202.627791039417\right]
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
(sin(x)x+2(δ(x)sign(x)x)sin(x)x2+2cos(x)sign(x)x2)=0- (\frac{\sin{\left(x \right)}}{\left|{x}\right|} + \frac{2 \left(\delta\left(x\right) - \frac{\operatorname{sign}{\left(x \right)}}{x}\right) \sin{\left(x \right)}}{x^{2}} + \frac{2 \cos{\left(x \right)} \operatorname{sign}{\left(x \right)}}{x^{2}}) = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
limx(sin(x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{\left|{x}\right|}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
limx(sin(x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{\left|{x}\right|}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x)/|x|, dividida por x con x->+oo y x ->-oo
limx(sin(x)xx)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \left|{x}\right|}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
limx(sin(x)xx)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \left|{x}\right|}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
sin(x)x=sin(x)x\frac{\sin{\left(x \right)}}{\left|{x}\right|} = - \frac{\sin{\left(x \right)}}{\left|{x}\right|}
- No
sin(x)x=sin(x)x\frac{\sin{\left(x \right)}}{\left|{x}\right|} = \frac{\sin{\left(x \right)}}{\left|{x}\right|}
- No
es decir, función
no es
par ni impar